Resolve the relative phase between quark-interchange and pion-exchange amplitudes

Determine the signed microscopic matching between the quark-interchange amplitudes and the chiral one-pion-exchange amplitudes, thereby fixing the unresolved global relative sign that controls constructive or destructive interference in the coherent $P_c\to\Lambda_c\bar D^{(*)}$ widths.

Background

The paper computes short-range quark-interchange and retarded one-pion-exchange amplitudes in independently defined conventions. Their relative global phase is not derived, so the authors introduce a correlated matching parameter $\eta_{\mathrm{match}}=\pm1$ and report both coherent-width branches.

The ambiguity is especially important for $P_c(4312)\to\Lambda_c\bar D^*$, where the pion-exchange amplitude is relatively large compared with the quark-interchange amplitude. Resolving the sign requires a signed microscopic matching of the quark-model states to the chiral heavy-hadron fields or an interference-sensitive observable.